Abstract

The dimensionless zero-frequency electronic first hyperpolarizability 3−1/4βE107/2m3/2(eℏ)−3 of an electron in one dimension was maximized by adjusting the shape of a piecewise linear potential. Careful maximizations converged quickly to 0.708951 with increasing numbers of parameters. The Hessian shows that β is strongly sensitive to only two parameters in the potential: sensitivity to additional parameters decreases rapidly. With more than two parameters, a wide range of potentials and an apparently narrower range of wavefunctions have nearly optimal hyperpolarizability. Modulations of the potential to which the unique maximum is insensitive were characterized. Prospects for concise description of the two important constraints on near-optimum potentials are discussed.

References

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Table 1.

For Different Numbers of Parameters in the Potential: (from left) Optimized Intrinsic Hyperpolarizabilities , the Ratio of Eigenvalues of the Hessian Matrix of the Intrinsic Hyperpolarizability, Oscillator Strengths Between Various States, the Ratio E of the Energy Level Spacings

Table 2.

Eigenvalues of Hessian for N=7 parameters in Each of Three Measures: (from left) the Numerically Natural Measure Induced by the Parametrization and then the Measures ρ0=|ψ0|2 and ρ10=|ψ0|2+|ψ1|2

Eigenvalues of Hessian

Derivatives of Dipole Matrix Elements

Derivative of Energy Ratio

NNM

ρ0

ρ10

x10′

Δx10′

x20′

x21′

E′

−3.28

−22.2

−8.10×10−1

−3.21×10−1

−2.15

−0.35

−0.57

−1.90

−3.14×10−1

−4.70

−1.76×10−1

−7.07×10−2

5.35×10−3

−5.46×10−2

−0.30

−0.77

−1.09×10−2

−1.15

−8.19×10−2

−2.05×10−3

4.96×10−3

1.61×10−2

2.80×10−2

−1.62×10−2

−4.83×10−3

−2.47×10−1

−3.16×10−2

−1.47×10−3

1.13×10−2

−2.80×10−2

−5.23×10−3

1.48×10−1

−2.58×10−4

−8.03×10−2

−5.54×10−3

−9.80×10−5

3.91×10−4

−2.13×10−3

−3.28×10−3

4.61×10−3

−1.66×10−5

−4.65×10−3

−3.03×10−3

−1.23×10−5

−7.71×10−6

6.72×10−4

8.49×10−4

−1.80×10−3

−1.04×10−6

−3.12×10−4

−2.02×10−4

3.44×10−7

−4.85×10−7

−4.26×10−7

−2.86×10−6

4.52×10−6

Derivatives of the dipole matrix elements and energy ratio in the direction of the eigenvector associated with each eigenvalue are also displayed.

Tables (2)

Table 1.

For Different Numbers of Parameters in the Potential: (from left) Optimized Intrinsic Hyperpolarizabilities , the Ratio of Eigenvalues of the Hessian Matrix of the Intrinsic Hyperpolarizability, Oscillator Strengths Between Various States, the Ratio E of the Energy Level Spacings